Are You Ready? Angle Relationships

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1 SKILL 5 Angle Relationships Teaching Skill 5 Objective Identify angle relationships. Begin by explaining to students that angle relationships often provide information about the measure of the angles. Point out that there are a number of angle postulates and theorems that establish congruence between certain types of angles. Emphasize that it is important to be able to identify angle relationships in order to apply those congruence postulates and theorems. Review the definitions and examples of adjacent angles, vertical angles, complementary angles, and supplementary angles. Point out the difference between complementary and supplementary angles. Ask: Which pair of angles form a straight angle? (supplementary) Instruct students to complete the practice exercises. PRACTICE ON YOUR OWN In exercises 4, students choose which description best fits the angle relationships. In exercises 5 8, students use a diagram to give examples of different types of angle relationships. CHECK Determine that students know how to identify angle relationships. Students who successfully complete the Practice on Your Own and Check are ready to move on to the next skill. COMMON ERRORS Students may confuse the definitions of complementary and supplementary. Students who made more than errors in the Practice on Your Own, or who were not successful in the Check section, may benefit from the Alternative Teaching Strategy. Alternative Teaching Strategy Objective Identify angle relationships. Materials needed: multiple enlarged copies of the game cards shown below 40 Supplement of 95 0 Complement of Angle adjacent to Larger Larger Smaller Angle PQR if P R angle PQS is Q 39 Smaller 80 4 S 4 Tell students they are going to play Larger, Smaller, or Equal. Before you begin, review the definitions and a few examples of adjacent, vertical, complementary, and supplementary angles. Then, give each student a set of shuffled game cards. Tell the students that when you say Go, they should match their cards according to the small numbers in the lower right corner of the card. Then students should determine which card represents the smaller angle and which represents the larger angle. They should place the smaller angle in a pile on their left and the larger angle on their right. If the two angles are equal, they should place them both in a center pile. The first student to correctly separate their cards wins. 6 Holt Geometry

2 Name Date Class SKILL 5 Angle Relationships Adjacent Angles Definition: two angles that share a side and a vertex, but no interior points Example: Angle Relationships Vertical Angles Definition: two angles whose sides are opposite rays 3 Example: 4 and 3 and 4 Complementary Angles Definition: two angles, the sum of whose measures is 90 Supplementary Angles Definition: two angles, the sum of whose measures is 80 Examples: 40 and 50 ; Examples: 60 and 0 ; Practice on Your Own Circle the better description for each labeled angle pair.. complementary angles supplementary angles. vertical angles supplementary angles 3. adjacent angles vertical angles and 55 complementary angles supplementary angles Use the diagram to the right to give an example of each angle pair. F E 5. adjacent angles 6. complementary angles 7. vertical angles 8. supplementary angles G A C D B Check Circle the better description for each labeled angle pair. 9. complementary angles adjacent angles 0. complementary angles adjacent angles Use the diagram to the right to give an example of each angle pair.. vertical angles. complementary angles S T P U 3. adjacent angles 4. supplementary angles R Q 6 Holt Geometry

3 SKILL 7 Classify Real Numbers Teaching Skill 7 Objective Classify real numbers. Explain to students that the set of real numbers includes several smaller classifications of numbers. Review each of the classifications. Ask: Which classification includes the fewest number of elements? (natural numbers) Why? (All the other classifications build on the natural number set. That is, they include all the natural numbers plus other numbers.) Have students review the different classifications, looking for the number 5. Ask? Is the number 5 a member of each of the sets of numbers? (Yes) Why is it a rational number? (The number 5 can be written as 5, 0, 0 4, etc.) Point out that all integers, and therefore all whole numbers and all natural numbers, can be written as rational numbers. Also point out that all terminating decimals can be written as rational numbers. Review the example: 0. or 0 5. PRACTICE ON YOUR OWN In exercises 9, students classify real numbers. CHECK Determine that students know how to classify real numbers. Students who successfully complete the Practice on Your Own and Check are ready to move on to the next skill. COMMON ERRORS Students may not understand the difference between whole numbers and natural numbers. Students who made more than errors in the Practice on Your Own, or who were not successful in the Check section, may benefit from the Alternative Teaching Strategy. Alternative Teaching Strategy Objective Classify real numbers. Materials needed: overhead transparency with the Venn Diagram shown below Real Numbers Rational Numbers Integers Whole Numbers Natural Numbers Explain to students that a Venn Diagram is a way of visually organizing sets of data that may overlap. Display the overhead transparency. Ask: Which set of numbers is the smallest? (natural numbers) Which set of numbers is the largest? (real numbers) Remind students that a natural number is also called a counting number. Ask? When you count, what number do you begin with? () Write the number in the Natural Numbers section of the Venn Diagram. Write the next few natural numbers in the section, followed by three dots: (, 3, 4, ) Point out that the section for Natural Numbers is inside Whole Numbers, which is inside Integers, which is inside Rational Numbers, which is inside Real Numbers. Ask: What does this mean with respect to the number? (It is a natural number, a whole number, an integer, a rational number, and a real number.) Work through the examples below, placing each one in the appropriate section of the diagram. Have students name all the classifications to which the numbers belong based on the way in which the sections of the Venn Diagram are nested. 0, 5, 4 5, 0.3, 7, 0 6, 4,., Holt Geometry

4 Name Date Class SKILL 7 Classify Real Numbers Natural Numbers The set of natural numbers includes all counting numbers. Whole Numbers The set of whole numbers includes all natural numbers and zero. {,, 3, 4, 5, 6, 7, 8, 9, 0, } {0,,, 3, 4, 5, 6, 7, 8, 9, 0, } Integers Rational Numbers The set of integers includes all positive and A rational number is any number that can negative whole numbers and zero. be written in the form a, where a and b are b integers and b is not equal to 0. {, 5, 4, 3,,, 0,,, 3, 4, 5, } Examples: 3, 0 4, 9, 0.5, 9., 7 3, etc. Practice on Your Own Tell if each number is a natural number, a whole number, an integer, or a rational number. Include all classifications that apply ,000,000 Check Tell if each number is a natural number, a whole number, an integer, or a rational number. Include all classifications that apply Holt Geometry

5 SKILL Points, Lines, and Planes Teaching Skill Objective Identify points, segments, rays, lines, and planes. Review with students the definitions provided on the worksheet of this lesson. Also review the notations. Ask: What is the difference between a segment and a ray? (A segment only has one endpoint, while a ray has two.) Ask: What is the difference between a ray and a line? (A ray has one endpoint while a line does not have any.) Review the example. Point out that there can be multiple rays or lines through any given point. Ask: Can you write segment BA instead of segment AB? (Yes) Can you write ray AB instead of ray BA? (No) Why not? (Because you must begin at the endpoint and proceed toward the arrow) Ask: Why are there no lines in the diagram? (Because the series of points only extend in one direction, making them rays instead of lines) PRACTICE ON YOUR OWN In exercises 0, students identify points, segments, rays, lines, and planes on a diagram. CHECK Determine that students know how to identify points, segments, rays, lines, and planes. Students who successfully complete the Practice on Your Own and Check are ready to move on to the next skill. COMMON ERRORS Students may confuse lines and rays or rays and segments because they do not pay attention to the arrows and endpoints. Students who made more than errors in the Practice on Your Own, or who were not successful in the Check section, may benefit from the Alternative Teaching Strategy. Alternative Teaching Strategy Objective Identify points, segments, rays, lines, and planes using a rectangular solid. Give students an enlarged copy of a rectangular solid as shown below. D A E H Remind students that a rectangular solid has faces, edges, and vertices. Give examples of each of these. For example, ABCD is a face; AD is an edge; and A is a vertex. Ask: Which of these is the same as a point? (a vertex) Which is the same as a segment? (an edge) Which is the same as a plane if it were extended in all directions? (a face) Have students count the number of vertices (8) and then name eight points. (A, B, C, D, E, F, G, and H ) Next have students count the number of edges () and name twelve segments. Remind students that segment AB is the same thing as segment BA. (AB, AD, AH, BG, BC, CD, CF, DE, EF, EH, FG, GH ) Ask: How could you change any of the segments (edges) to make them rays? (You could extend ONE of the sides without end.) How could you change any of the segments to make them lines? (You could extend both of the sides without end.) Finally, have students count the number of faces (6) and then name six planes. (ABCD, ABGH, ADEH, BCFG, CDEF, and EFGH ) Repeat this exercise by pointing to the walls, edges, and corners of the classroom and having students identify them as points, segments, or planes. C B F G 55 Holt Geometry

6 Name Date Class SKILL Points, Lines, and Planes Object Definition Notation Point A location in space that has no size. A Segment Ray Line Plane A part of a line consisting of two endpoints and all the points between. A part of a line consisting of one endpoint and all the points of the line on one side of the endpoint. A series of points that extends without end in opposite directions. A flat (two-dimensional) surface that extends without end in all directions. A plane has no thickness. Example: Use the diagram to name a point, a segment, a ray, a line, and a plane (if possible). B A C D Answers Any one of the following are points: A, B, C, or D. There are two segments: BA _ and BC. There are two rays: BA and BC. There are no lines. There is one plane: _ AB AB AB Practice on Your Own Use the diagram to the right to name each of the following.. a point. a line 3. a segment 4. a ray 5. a plane 6. a ray with endpoint O 7. the line that passes through the point M 8. the plane that contains the point Q M O N Q P 9. a segment with one endpoint at point O 0. the point where lines MP and ON meet Check Use the diagram to the right to name each of the following.. two points and. a segment 3. two rays and 4. a line 5. two planes and 6. the plane that contains line CD 56 Holt Geometry A D C B E

7 SKILL 68 Solve One-Step Equations Teaching Skill 68 Objective Solve one-step equations. Explain to students that inverse operations are operations that undo each other. Direction students attention to the first example. Ask a volunteer to read the addition equation x 5 5. Ask: What operation is being done to the variable? (5 is being added to it.) How can you undo this? (Subtract 5.) So, what is the inverse operation of addition? (subtraction) Work through the next example. Stress that since subtraction is the inverse operation of addition, addition is the inverse operation of subtraction. Go through a similar process to explain the multiplication and division examples. Remind students that they should be careful when working with negatives. Ask: If a multiplication equation contains x, what do you divide by? () Does that mean the answer is negative? (not necessarily) Give examples if time permits. Have students complete the practice exercises. PRACTICE ON YOUR OWN In exercises, students solve one-step equations that require addition, subtraction, multiplication, or division. CHECK Determine that students know how to solve one-step equations. Students who successfully complete the Practice on Your Own and Check are ready to move on to the next skill. COMMON ERRORS Students may forget to use an inverse operation when solving equations. Students who made more than errors in the Practice on Your Own, or who were not successful in the Check section, may benefit from the Alternative Teaching Strategy. Alternative Teaching Strategy Objective Solve one-step equations. Materials needed: multiple copies of the flashcards shown below (index cards work nicely) n 3 9 n 4 n 6 n 8 n 5 n 5 5n 50 7n 4 9n 45 n 6 4 n 5 n 00 5 Tell students they are going to practice identifying inverse operations and then solving one-step equations. Remind students that inverse operations undo each other. Have students work in pairs. Give each pair of students one set of equation cards. Instruct students to mix the cards up and leave them face down. Students should take turns drawing one card at a time. The student who draws the card should face it toward their partner. The partner should read the equation aloud, identify the operation being done to the variable, and then identify how to undo the operation. The partner does not need to solve the equation. The student holding the card should then turn the card around and read it, confirm their partner s answers, and then solve the equation. Students should repeat the exercise until all the cards have been drawn and all the equations solved. An extension of this exercise is to have students make their own index cards which should include addition, subtraction, multiplication and division equations. 47 Holt Geometry

8 Name Date Class SKILL 68 Solve One-Step Equations To solve a one-step equation, do the inverse of whatever operation is being done to the variable. Remember, because it is an equation, what is done to one side of the equation must also be done to the other side. Solve an addition equation using subtraction. x x 0 Solve a multiplication equation using division. 7x 4 7x x 6 Solve a subtraction equation using addition. x x 5 Solve a division equation using multiplication. x 3 x 3 x 36 Practice on Your Own Solve.. m 5 9. h x b y 6. k p 7 8. t x 0. 5 h 6. x 4. r 9 4 Check Solve. 3. 3x 5 4. c 5. d s z 7 8. w b 0 0. x Holt Geometry